I'm reading Serge Lang's Fundamentals of Differential Geometry. In chapter VIII section 1, he defines a covariant derivative as follows:
By a covariant derivative $D$ we mean an $\bf{R}$-bilinear map $$D:\Gamma T(X)\times\Gamma T(X)\to\Gamma T(X),$$ denoted by $(\xi,\eta)\mapsto D_\xi\eta$, satisfying the two conditions:
COVD 1. (a) In the first variable $\xi$, $D_\xi\eta$ is Fu-linear. (b) For a function $\varphi$, define $D_\xi\varphi=\xi\phi=\mathscr{L}_\xi\varphi$ to be the Lie derivative of the function. Then in the second variable $\eta$, $D_\xi\eta$ is a derivation. Thus (a) and (b) can be written in the form: $$D_{\varphi\xi}\eta=\varphi D_\xi\eta\quad\text{and}\quad D_\xi(\varphi\eta)=(D_\xi\varphi)\eta+\varphi D_\xi\eta.$$ COVD 2. $D_\xi\eta-D_\eta\xi=[\xi,\eta].$
He then goes on:
Having defined $D_\xi$ on functions and vector fields, we may extend the definition to all differential forms, or even to multilinear tensor fields. Let $\omega$ be in $\Gamma L^r(T(X))$, i.e. $\omega$ is a multilinear tensor field on $X$, not necessarily alternating. We define $D_\xi\omega$ by giving its value on vector fields $\eta_1,\dots,\eta_r$, namely \begin{equation*} (D_\xi\omega)(\eta_1,\dots,\eta_r)=\mathscr{L}_\xi(\omega(\eta_1,\dots,\eta_r))- \sum_{j=1}^r\omega(\eta_1,\dots,D_\xi\eta_j,\dots,\eta_r). \end{equation*}
Now I get what he is trying to do, but I don't see how you can get away with defining a multilinear tensor field globally by defining it on global vector fields. (By the way, in the introduction to this chapter, $X$ is just a plain vanilla $C^\infty$ manifold, which for Lang means modeled on some set of Banach spaces. $\Gamma T(X)$ is the $\mathbf{R}$-vector space of vector fields, which happens to also be a module over the ring of functions Fu$(X)$. And finally, $\pi:TX\to X$ is the natural map of the tangent bundle to $X$.)
In (my take on) Lang's formalism, $\omega\colon X\to L^r_X(TX)$ where the fiber over $x$ is $(L^r_X(TX))_x=\{x\}\times L^r(T_xX)$. Furthermore, if $(V,\psi)$ is a chart for $X$, with $\psi\colon V\to\mathbf{E}$, then we have a trivializing map $\tau\colon\pi^{-1}(V)\to V\times\mathbf{E}$ for $\pi$ and a trivializing map $\gamma\colon L^r_X(\pi)^{-1}(V)\to V\times L^r(\mathbf{E})$ for the vector bundle $L^r_X(\pi)\colon L^r_X(TX)\to X$. To be a section, $\omega(x)$ needs to be in $L^r_X(\pi)^{-1}(x)$, so if $x\in V$, then with $\gamma_x=\text{pr}_2\circ\gamma|L^r_X(\pi)^{-1}(x)$, we need $\gamma_x(\omega(x))\in L^r(\mathbf{E})$. So if we are defining $D_\xi\omega$, we need $\gamma_x(D_\xi\omega(x))\in L^r(\mathbf{E})$.
Now, on page 124 (Chapter V section 3) Lang indicates that for vector fields $\xi_1,\dots,\xi_r$ defined on an open subset $U$ of $X$, $\langle\omega,\xi_1\times\dots\times\xi_r\rangle$ is a function on $U$, whose value at $x\in U$ is $\omega(x)(\xi_1(x),\dots,\xi_r(x))$. In my take on Lang's formalism this would actually equal $(\gamma_x(\omega(x)))(\tau_x(\xi_1(x)),\dots,\tau_x(\xi_r(x)))$, where $\tau_x=\text{pr}_2\circ\tau|\pi^{-1}(x)$. [I admit that I am glossing over the fact that $\xi_i$ is a vector field on $U$, which means it maps to $TU$. For a modicum of simplicity, I'm just assuming that $\xi_i$ maps $U$ to $TX$, which I don't think will hurt.] So I assume this is what Lang means by $(D_\xi\omega)(\eta_1,\dots,\eta_r)$, except that it seems we have to assume that $\eta_j$ is a vector field on $X$, since Lang has not given a definition for $D_\xi\eta_j$ when $\xi\in\Gamma T(X)$ but $\eta_j\in\Gamma T(U)$. But since $\gamma_x(\omega(x))\in L^r(\mathbf{E})$, we need to specify the value of $\gamma_x(\omega(x))$ on every $(v_1,\dots,v_r)\in\mathbf{E}^r$. And that would mean that given $v_j\in\mathbf{E}$, we would need to be able to exhibit a global vector field $\xi_j$ such that $\tau_x(\xi_j(x))=v_j$. I know how to make a local vector field $\xi^V_j\colon V\to TV$ or even mapping to $TX$ which sends $x$ to $\tau^{-1}(x,v_j)$, but I don't know how to globalize it to all of $X$.
Am I missing something? Is there a way to define a vector field $\xi:X\to TX$ such that its local representation in a chart $(V,\psi)$ at a given point $\psi x$ is $v\in\mathbf{E}$? Do I have to assume something like $X$ admitting partitions of unity subordinate to a particular open covering in order to do so? If that is the case, how do you use the partition of unity to do it?